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feat(Analysis/Normed/Module/WeakDual): register bornology and boundedness API (leanprover-community#36332)
## Description This PR registers `Bornology` instance and introduces a boundedness API as discussed [on Zulip](https://leanprover.zulipchat.com/#narrow/channel/287929-mathlib4/topic/instBornology.20for.20WeakSpace.20and.20WeakDual/with/577457935). ### Main additions - **`WeakDual.instBornology`**: Bornology instance on `WeakDual 𝕜 E` inherited from `StrongDual 𝕜 E` via `inferInstanceAs`. This makes `IsBounded` available directly on weak dual sets, where boundedness means operator-norm boundedness. - **`isBounded_iff_isVonNBounded`**: For normed spaces over `ℝ` or `ℂ`, norm-boundedness and weak-* von Neumann boundedness coincide (via Banach–Steinhaus). - **`isBounded_polar`** and **`isBounded_closedBall`**: Standalone lemmas replacing repeated inline `isBounded_toStrongDual_preimage_iff_isBounded.mpr ...` patterns. - **Simp lemmas**: `isBounded_toStrongDual_preimage_iff_isBounded` and `isBounded_toWeakDual_preimage_iff_isBounded` for translating between weak and strong dual boundedness. ### API simplification The existing lemmas `isClosed_image_coe_of_bounded_of_closed`, `isCompact_of_bounded_of_closed`, and `isSeqCompact_of_isBounded_of_isClosed` now take `IsBounded s` directly instead of the more verbose `IsBounded (StrongDual.toWeakDual ⁻¹' s)`, since the two are definitionally equal with the new bornology instance. ### Minor changes - Replaced `grind` tactic usage with explicit term - Fixed deprecation dates from 2024-11-19 to 2025-11-19 - Renamed local variable `F` to `E` for consistency with the rest of the file - Added comprehensive module and declaration docstrings
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Mathlib/Analysis/Normed/Algebra/Basic.lean

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@@ -62,7 +62,7 @@ instance [ProperSpace 𝕜] : CompactSpace (characterSpace 𝕜 A) := by
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intro φ hφ
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rw [Set.mem_preimage, mem_closedBall_zero_iff]
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exact (norm_le_norm_one ⟨φ, ⟨hφ.1, hφ.2⟩⟩ :)
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exact (isCompact_closedBall 𝕜 0 _).of_isClosed_subset CharacterSpace.isClosed h
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exact (isCompact_closedBall 0 _).of_isClosed_subset CharacterSpace.isClosed h
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end CharacterSpace
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